#143166

order	512
nilpotency class	4
isoclinism	isoclinic class of #143166
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10018
commutator, Frattini, cosocle	C₄×M₁₆	C₂³
LCS	G ▷ C₄×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ G
LECS	G ▷ C₄×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

392 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2)(3,13)(4,18)(5,10)(6,11)(7,12)(8,15)(9,21)(14,29)(16,26)(17,30)(19,23)(20,28)(22,27)(24,32)(25,31);(1,3)(2,8)(4,19)(5,14)(6,16)(7,17)(9,26)(10,23)(11,24)(12,25)(13,18)(15,21)(20,31)(22,32)(27,30)(28,29);(3,14)(4,20)(5,6)(8,23)(9,27)(10,11)(15,31)(16,17)(18,28)(21,22)(24,25)(26,29)

Polycyclic presentation:

2^9|d^2=gi,e^2=gh,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,f^b=fi,g^b=gi,d^c=dh,e^c=egh,f^c=fi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339885
maximal normal subgroup of	#72339885

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50362);